ch 7.3 volumes calculus graphical, numerical, algebraic by finney, demana, waits, kennedy
TRANSCRIPT
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Ch 7.3 VolumesCalculus Graphical, Numerical, Algebraic byFinney, Demana, Waits, Kennedy
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Volume of a Solid
b
a
x
x
The definition of a solid of known integrable cross section
area A from x = 0 to x = b is the integral of A from a to b,
V = A dx.
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, the area of the cross section.
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xy
A 45o wedge is cut from a cylinder of radius 3 as shown.Find the volume of the wedge.
You could slice this wedge shape several ways, but the simplest cross section is a rectangle.
If we let h equal the height of the slice then the volume of the slice is: 2V x y h dx
Since the wedge is cut at a 45o angle:x
h45o h x
Since2 2 9x y 29y x
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xy
2V x y h dx h x
29y x
22 9V x x x dx
3 2
02 9V x x dx
29u x 2 du x dx
0 9u 3 0u
10
2
9V u du
93
2
0
2
3u
227
3 18
Even though we started with a cylinder, does not enter the calculation!
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Cavalieri’s Theorem:
Two solids with equal altitudes and identical parallel cross sections have the same volume.
Identical Cross Sections
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2
0
0
0
2
V sin2
11 cos 2x
2 2
sin
4 2
4
4
= x dx
= dx
2x = x +
=
=
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y x Suppose I start with this curve.
My boss at the ACME Rocket Company has assigned me to build a nose cone in this shape.
So I put a piece of wood in a lathe and turn it to a shape to match the curve.
Disk Method
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y xHow could we find the volume of the cone?
One way would be to cut it into a series of thin slices (flat cylinders) and add their volumes.
The volume of each flat cylinder (disk) is:
2 the thicknessr
In this case:
r= the y value of the function
thickness = a small change
in x = dx
2
x dx
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y xThe volume of each flat cylinder (disk) is:
2 the thicknessr
If we add the volumes, we get:
24
0x dx
4
0 x dx
42
02x
8
2
x dx
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This application of the method of slicing is called the disk method. The shape of the slice is a disk, so we use the formula for the area of a circle to find the volume of the disk.
If the shape is rotated about the x-axis, then the formula is:
2 b
aV y dx
Since we will be using the disk method to rotate shapes about other lines besides the x-axis, we will not have this formula on the formula quizzes.
2 b
aV x dy A shape rotated about the y-axis would be:
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The region between the curve , and the
y-axis is revolved about the y-axis. Find the volume.
1x
y 1 4y
y x
1 1
2
3
4
1.707
2
1.577
3
1
2
We use a horizontal disk.
dy
The thickness is dy.
The radius is the x value of the function .1
y
24
1
1 V dy
y
volume of disk
4
1
1 dy
y
4
1ln y ln 4 ln1
02ln 2 2 ln 2
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The natural draft cooling tower shown at left is about 500 feet high and its shape can be approximated by the graph of this equation revolved about the y-axis:
2.000574 .439 185x y y x
y
500 ft
500 22
0.000574 .439 185 y y dy
The volume can be calculated using the disk method with a horizontal disk.
324,700,000 ft
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Disks Example
The region between the graph of f(x) = 2 + x cos x and the x axis over the interval [-2,2] is revolved about the x-axis to generate a solid. Find the volume of the solid
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Circular Cross Sections
The region between the graph of f(x) = 2 + x cos x and the x axis over the interval [-2,2] is revolved about the x-axis to generate a solid. Find the volume of the solid
Area of the cross section =
The volume of the solid is:
2A(x) = f x
2 2 2 3
-2 -2V = A(x) dx = 2 + x cos x dx = 52.429 units
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End of Ch 7.3 Day 1
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Ch 7.3 Day 2: Washer Method
3
2
1
-2 2
h x = -3x
g x = 3x
f x = x2
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The region bounded by and is revolved about the y-axis.Find the volume.
2y x 2y x
The “disk” now has a hole in it, making it a “washer”.
If we use a horizontal slice:
The volume of the washer is: 2 2 thicknessR r
2 2R r dy
outerradius
innerradius
2y x
2
yx
2y x
y x
2y x
2y x
2
24
0 2
yV y dy
4 2
0
1
4V y y dy
4 2
0
1
4V y y dy
42 3
0
1 1
2 12y y
168
3
8
3
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This application of the method of slicing is called the washer method. The shape of the slice is a circle with a hole in it, so we subtract the area of the inner circle from the area of the outer circle.
The washer method formula is: 2 2 b
aV R r dx
Like the disk method, this formula will not be on the formula quizzes. I want you to understand the formula.
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2y xIf the same region is rotated about the line x=2:
2y x
The outer radius is:
22
yR
R
The inner radius is:
2r y
r
2y x
2
yx
2y x
y x
4 2 2
0V R r dy
2
24
02 2
2
yy dy
24
04 2 4 4
4
yy y y dy
24
04 2 4 4
4
yy y y dy
14 2 2
0
13 4
4y y y dy
432 3 2
0
3 1 8
2 12 3y y y
16 64
243 3
8
3
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Washer Cross Section
The region in the first quadrant enclosed by the y-axis and the graphs of y = cos x and y = sin x is revolved about the x-axis to form a solid. Find its volume.
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Washer Cross Section
The region in the first quadrant enclosed by the y-axis and the graphs of y = cos x and y = sin x is revolved about the x-axis to form a solid. Find its volume.
4 4 2 2
0 0
4
0
43
0
V = A(x) dx = cos x - sin x dx
= cos 2x dx
sin 2x = = units
2 2
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Volumes of Solids: End of Day 2
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Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2006
7.3 Day 3
The Shell Method
Japanese Spider CrabGeorgia Aquarium, AtlantaGrows to over 12 feet wide
and lives 100 years.
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Find the volume of the region bounded by , , and revolved about the y-axis.
2 1y x 2x 1y2 1y x
We can use the washer method if we split it into two parts:
25 2 2
12 1 2 1y dy
21y x 1x y
outerradius
innerradius
thicknessof slice
cylinder
5
14 1 4y dy
5
15 4y dy
52
1
15 4
2y y
25 125 5 4
2 2
25 94
2 2
164
2
8 4 12
Japanese Spider CrabGeorgia Aquarium, Atlanta
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If we take a vertical slice and revolve it about the y-axis
we get a cylinder.
cross section
If we add all of the cylinders together, we can reconstruct the original object.
2 1y x
Here is another way we could approach this problem:
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cross section
The volume of a thin, hollow cylinder is given by:
Lateral surface area of cylinder thickness
=2 thicknessr h r is the x value of the function.
circumference height thickness
h is the y value of the function.
thickness is dx. 2=2 1 x x dx
r hthicknesscircumference
2 1y x
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cross section
=2 thicknessr h
2=2 1 x x dx
r hthicknesscircumference
If we add all the cylinders from the smallest to the largest:
2 2
02 1 x x dx
2 3
02 x x dx
24 2
0
1 12
4 2x x
2 4 2
12
This is called the shell method because we use cylindrical shells.
2 1y x
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2410 16
9y x x
Find the volume generated when this shape is revolved about the y axis.
We can’t solve for x, so we can’t use a horizontal slice directly.
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2410 16
9y x x Shell method:
Lateral surface area of cylinder
=circumference height
=2 r h Volume of thin cylinder 2 r h dx
If we take a vertical sliceand revolve it about the y-axiswe get a cylinder.
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2410 16
9y x x Volume of thin cylinder 2 r h dx
8 2
2
42 10 16
9x x x dx r
h thickness
160
3502.655 cm
Note: When entering this into the calculator, be sure to enter the multiplication symbol before the parenthesis.
circumference
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When the strip is parallel to the axis of rotation, use the shell method.
When the strip is perpendicular to the axis of rotation, use the washer method.
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Volumes Using Cylindrical Shells
The region bounded by the curve y = , the x-axis, and the line x = 4 is revolved about the x-axis to generate a solid. Find the volume of the solid using cylindrical shells.
x
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Volumes Using Cylindrical Shells
The region bounded by the curve y = , the x-axis, and the line x = 4 is revolved about the x-axis to generate a solid. Find the volume of the solid using cylindrical shells.
x
2 2
0V = 2 y 4 - y dy
= 8
2
2
Radius = y
x = y
Shell height = 4 - y